Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i} - \hat{j} + \hat{k}$,$\hat{i} + 2\hat{j} + \mu \hat{k}$ and $3\hat{i} - 4\hat{j} + 5\hat{k}$,where $\lambda - \mu = 5$,are coplanar,then $\sum_{(\lambda, \mu) \in S} 80(\lambda^2 + \mu^2)$ is equal to :

If $\vec a = 3\vec j + 4\vec k$,$\vec b = 2\vec i + \vec k$ and $\vec c$,$\vec d$ are respectively the components of $\vec a$ parallel and perpendicular to $\vec b$,then the value of the scalar triple product $\left[ {(\vec a \times \vec c) \times (\vec c \times \vec d), (\vec c \times \vec d) \times (\vec d \times \vec a), (\vec d \times \vec a) \times (\vec a \times \vec c)} \right]$ is equal to:

$ [\vec{a}+2 \vec{b}-\vec{c}, \vec{a}-\vec{b}, \vec{a}-\vec{b}-\vec{c}] $

If $A$,$B$,and $C$ are three non-coplanar vectors,then $(A + B + C) \cdot ((A + B) \times (A + C)) = \dots$

If $a, b, c$ are non-negative distinct numbers and $a \hat{\imath}+a \hat{\jmath}+c \hat{k}$,$\hat{\imath}+\hat{k}$ and $c \hat{\imath}+c \hat{\jmath}+b \hat{k}$ are coplanar vectors,then

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is